Proposition: Mixed-term identification

For every smooth $u$,

Thus the map on the first bare insertion is exactly the mixed term of the classical relational expansion, with its minus sign.

Proof. Theorem thm:retarded identifies the left side with $⟨c_1[h],{\rho }_{1,g}[h;u]⟩$. Use $c_1[h]={\partial }_{\mu }(f{w}^{\mu }[h])$ and integrate by parts.